%I #29 Jun 01 2019 11:31:26

%S 0,0,1,1,2,2,2,2,3,3,4,4,2,4,3,3,4,3,2,4,2,4,5,1,3,3,2,5,4,3,6,2,4,4,

%T 4,7,4,3,3,6,7,7,3,5,3,6,7,5,7,4,4,4,5,6,7,4,4,6,6,6,6,3,6,6,6,8,7,5,

%U 3,4,6,8,4,3,4,3,6,6,4,5,6,4,6,6,9,7,4,5,8,9,6,5,5,7,5,6,2,7,6,5

%N Number of ways to write n as 6^i + 3^j + A008347(k), where i, j and k > 0 are nonnegative integers.

%C Conjecture 1: a(n) > 0 for all n > 2. In other words, each n = 3,4,... can be written as 6^i + 3^j + prime(k) - prime(k-1) + ... + (-1)^(k-1)*prime(1), where i, j and k > 0 are nonnegative integers.

%C Conjecture 2: If {a,b} is among {2,m} (m = 3..14), {3,4}, {3,5}, then any integer n > 2 can be written as a^i + b^j + A008347(k) with i, j and k > 0 nonnegative integers.

%C Using Qing-Hu Hou's program, we have verified Conjectures 1 and 2 for n up to 10^9 and 10^7 respectively. - _Zhi-Wei Sun_, May 28 2019

%C Conjecture 1 verified up to 10^10. Conjecture 2 holds up to 10^10 for all cases except {2, 12} since 4551086841 cannot be written as 2^i + 12^j + A008347(k). - _Giovanni Resta_, May 28 2019

%H Zhi-Wei Sun, <a href="/A308403/b308403.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.

%e a(3) = 1 with 3 - (6^0 + 3^0) = 1 = A008347(2).

%e a(4) = 1 with 4 - (6^0 + 3^0) = 2 = A008347(1).

%e a(24) = 1 with 24 - (6^0 + 3^0) = 22 = A008347(13).

%e a(234) = 1 with 234 - (6^1 + 3^3) = 201 = A008347(90).

%e a(1134) = 1 with 1134 - (6^2 + 3^0) = 1097 = A008347(322).

%e a(4330) = 1 with 4330 - (6^3 + 3^0) = 4113 = A008347(1016).

%e a(5619) = 1 with 5619 - (6^1 + 3^3) = 5586 = A008347(1379).

%e a(6128) = 1 with 6128 - (6^0 + 3^0) = 6126 = A008347(1499).

%e a(16161) = 1 with 16161 - (6^3 + 3^0) = 15944 = A008347(3445).

%e a(133544) = 1 with 133544 - (6^0 + 3^8) = 126982 = A008347(22579).

%t Pow[n_]:=Pow[n]=n>0&&IntegerQ[Log[3,n]];

%t s[0]=0;s[n_]:=s[n]=Prime[n]-s[n-1];

%t tab={};Do[r=0;Do[If[s[k]>=n,Goto[bb]];Do[If[Pow[n-s[k]-6^m],r=r+1],{m,0,Log[6,n-s[k]]}];Label[bb],{k,1,2n-1}];tab=Append[tab,r],{n,1,100}];Print[tab]

%Y Cf. A000040, A000244, A000400, A008347, A303656, A303821, A308411.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, May 25 2019